User:Virginia-American/Sandbox/Ramanujan sum

Sums

Let

$T_{q}(n)=c_{q}(1)+c_{q}(2)+\dots +c_{q}(n){\mbox{ and }}$$U_{q}(n)=T_{q}+{\tfrac {1}{2}}\phi (q).$

Then

$\sigma _{-s}(1)+\sigma _{-s}(2)+\dots +\sigma _{-s}(n)$

$=\zeta (s+1)\left(n+{\frac {T_{2}(n)}{2^{s+1}}}+{\frac {T_{3}(n)}{3^{s+1}}}+{\frac {T_{4}(n)}{4^{s+1}}}+\dots \right)$

$=\zeta (s+1)\left(n+{\tfrac {1}{2}}+{\frac {U_{2}(n)}{2^{s+1}}}+{\frac {U_{3}(n)}{3^{s+1}}}+{\frac {U_{4}(n)}{4^{s+1}}}+\dots \right)-{\tfrac {1}{2}}\zeta (s),$

$d(1)+d(2)+\dots +d(n)$

$=-{\frac {T_{2}(n)\log 2}{2}}-{\frac {T_{3}(n)\log 3}{3}}-{\frac {T_{4}(n)\log 4}{4}}-\dots ,$

$d(1)\log 1+d(2)\log 2+\dots +d(n)\log n$

$=-{\frac {T_{2}(n)(2\gamma \log 2-\log ^{2}2)}{2}}-{\frac {T_{3}(n)(2\gamma \log 3-\log ^{2}3)}{3}}-{\frac {T_{4}(n)(2\gamma \log 4-\log ^{2}4)}{4}}-\dots ,$

$r_{2}(1)+r_{2}(2)+\dots +r_{2}(n)$

$=\pi \left(n-{\frac {T_{3}(n)}{3}}+{\frac {T_{5}(n)}{5}}-{\frac {T_{7}(n)}{7}}+\dots \right).$